#polyhedra

In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. In this short tutorial video, I am modeling an excavated dodecahedron in the Rhinoceros software. I use basic drawing and modeling commands. I aim to introduce this skill to beginner-level architects and designers. As the name suggests, this process includes the construction of the dodecahedron first. Then, the pentagonal faces of this polyhedron were “excavated” to the…

A truncated cuboctahedron is an Archimedean solid, meaning it is a highly symmetrical, convex polyhedron with identical vertices and faces of regular polygons. As its name suggests, you can create the truncated cuboctahedron by truncating (cutting off) the vertices of a cuboctahedron. A cuboctahedron is a rectified cube. After you model it, you can divide all its edges into 1/3s. Then, connect those division points to draw the squares, hexagons, and octagons. In this short tutorial video, I am folding…

An icosidodecahedron is a polyhedron that is part of the family of Archimedean solids. It has 32 faces, consisting of 12 regular pentagonal faces of a dodecahedron and 20 regular triangular faces of an icosahedron. It is highly symmetrical, with the same arrangement of faces around each vertex. In this short tutorial video, I am modeling and unrolling an icosidodecahedron. This polyhedron is very interesting and is mostly used as a basis for more sophisticated designs. Its dual polyhedron is a rhombic triacontahedron,…

Stellation of polyhedra refers to the process of extending the faces (or edges) of a polyhedron outward to form new, more complex shapes. These new shapes, called stellations, are created by adding extra vertices, edges, and faces that “project” outward from the original polyhedron’s structure. You can create an infinite number of new polyhedral forms, depending on how you extend the faces and edges. In this short tutorial video, I am modeling a stellated icosahedron. I use basic modeling commands…

A dual polyhedron is a concept in geometry where two polyhedra are related in such a way that the vertices of one polyhedron correspond to the faces of the other, and the faces of the first polyhedron correspond to the vertices of the second. The process of creating a dual polyhedron is called duality, and it applies to many regular, semi-regular, and some irregular polyhedra. In the case of Platonic solids, the cube and octahedron are duals of each other….

A Buckyball, (or Fullerene), is a molecule of 60 carbon atoms with a structure resembling a soccer ball. It is a specific type of fullerene, often referred to as C60, where the carbon atoms form a series of pentagons and hexagons in a pattern that creates a spherical shape. This structure is similar to the arrangement found in a truncated icosahedron, with 12 pentagonal and 20 hexagonal faces. The buckyball is named after architect Buckminster Fuller, who popularized geodesic domes,…

Truncation in mathematics and geometry refers to cutting off parts of a shape, typically the vertices, edges, or other extremities, to create a new modified figure. This process is often used with polygons, polyhedra, or functions. A truncated tetrahedron is a polyhedron we can derive from a regular tetrahedron by truncating (cutting off) its four vertices. This process creates a new shape with additional faces while maintaining its symmetry. In this tutorial video, I am modeling and unrolling a truncated…

A cube is a three-dimensional shape with six square faces, twelve edges, and eight vertices. It is one of the most familiar geometric solids and is a special type of rectangular prism where all sides are of equal length. A cube has a high degree of symmetry. Angles between its faces are right angles. It is often referred to as a “regular hexahedron”. Also cube is one of the Platonic solids, known for its perfect symmetry and simplicity. Nevertheless, the…

A dodecahedron is a three-dimensional polyhedron with twelve flat, regular pentagonal faces, twenty vertices, and thirty edges. It is one of the Platonic solids and is highly symmetrical, with each face being a regular pentagon. The dodecahedron’s shape is unique among the Platonic solids because its faces are polygons with five sides, unlike the others which have triangular faces. Due to its symmetry, the dodecahedron has been used in various applications, including as a model in geometry and in certain…

An icosahedron is a three-dimensional polyhedron with twenty triangular faces, twelve vertices, and thirty edges. It is one of the five Platonic solids and is highly symmetrical, with all faces being equilateral triangles. A regular icosahedron has equal edge lengths and angles between its faces, making it one of the most symmetrical shapes in three-dimensional space. In this short tutorial video, I am modeling and unrolling an icosahedron. I studied this beautiful solid here, here, and here before. The Icosahedron…